We know that $0<\dfrac{2^n}{3^n(n+7)} < \left(\dfrac{2}{3}\right)^n$ for any $n\ge 1$. Considering this fact, what does the direct comparison test say about $\sum\limits_{n=1}^{\infty }\dfrac{2^n}{3^n(n+7)}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A The series converges. (Choice B) B The series diverges. (Choice C) C The test is inconclusive.
Explanation: $\sum\limits_{n=1}^{\infty }~\left(\dfrac{2}{3}\right)^n$ is a geometric series with $~r=\dfrac{2}{3}~$, so it converges. Our given series is term-by-term less than a convergent series, so it converges as well.